FORMULATION AND CAPTURE OF THE SECOND VOLTAGE PEAK
OCCURING FROM PIEZOELECTRIC ENERGY HARVESTING USING A
PULSED RESONANT CONVERTER (PRC)
Alex Phipps and Toshikazu Nishida
Electrical and Computer Engineering, University of Florida, USA
Abstract: This paper examines the occurrence of the second voltage peak caused by the pulsed resonant converter
(PRC) and investigates the potential advantages of harvesting energy from this additional peak. Analytical
expressions are derived which describe the behavior of the second peak and these results are compared to SPICE
simulation in order to validate the theoretical models. It is shown that an optimal design point exists when
harvesting from both peaks ( ωτ = 1.76) which is different than the optimal point using only the standard primary
peak method ( ωτ = 1.53).
Keywords: piezoelectric, energy harvesting, pulsed resonant converter
source (representing an oscillating vibration source) in
parallel with the piezoelectric capacitance, Cpiezo, and
a parallel leakage term, Rpiezo. This model is typically
valid if the electromechanical coupling of the
transducer is low. The load of the PRC is a
rechargeable battery or super-capacitor.
INTRODUCTION
As the number of wireless and portable
electronics used in everyday life has increased,
interest has grown exponentially in methods that
extend the functional lifetime of these systems beyond
what batteries alone can provide. Vibration energy
harvesting, whereby mechanical vibrations are
converted into electrical energy, offers the potential to
extend the operational lifetime of these systems
indefinitely. Piezoelectric transduction schemes for
converting mechanical vibrations into electrical
energy are attractive because of their ability to
provide moderate power levels (~100uW) at voltages
necessary to operate standard electronics (2-10V).
Power converter circuitry is an essential and often
overlooked aspect in the design of energy harvesting
systems. The role of the power converter is to
condition the energy signals and to maximize the
amount of harvested power that is delivered to the
load. Various power conversion methodologies have
been examined for piezoelectric energy harvesting
including the rectifier-capacitor [1, 2], synchronized
switch harvesting on inductor (SSHI) [3, 4], and
synchronized charge extraction (SCE). Of these
methodologies, SCE circuits possess the advantage of
operating independently of their load and therefore
not requiring additional DC-DC conversion and the
associated inductive components. The pulsed resonant
converter (PRC) is a SCE topology that utilizes a
single, minimally-sized inductor and two switches to
extract energy from the piezoelectric transducer.
A schematic model of a PRC-based piezoelectric
energy harvesting system is shown in Fig. 1. The
input to the PRC is the piezoelectric transducer which
converts mechanical energy into electrical energy.
The transducer is modeled as a sinusoidal current
0-9743611-5-1/PMEMS2009/$20©2009TRF
LPRC
ipiezo(t) = R
Piezo CPiezo
Ipiezosin(ω t)
+
D1
D3 iLPRC(t)
PSwitch
VLoad
Vpiezo(t)
NSwitch
-
D2
D4
Fig. 1: Circuit representation of the transducer and
PRC.
For standard operation, the PRC transfers energy
accumulated on the piezoelectric capacitance to the
load when the magnitude of the piezoelectric voltage
is at a maximum, as shown in Fig. 2. The rectifier
allows energy to be harvested from this primary
voltage peak twice every vibration period. However,
as a result of the non-linear switching behavior of the
PRC, a secondary voltage wave is created which can
be harvested at its peak to provide additional energy
to the system. Previous works employing the PRC
have focused primarily on its circuit implementation
[5, 6]. These works show the occurrence of the
secondary wave in both simulation [5] and in
experimental results [6], but have not attempted to
describe or model this behavior.
In order to effectively design and optimize PRCbased energy harvesting systems for maximum power
collection, accurate modeling of the secondary wave
427
PowerMEMS 2009, Washington DC, USA, December 1-4, 2009
For an ideal PRC with zero losses, the power
delivered to the load is equal to the energy stored on
Cpiezo at the end of phase 1 given by
is necessary. This paper presents a detailed
examination of the secondary wave behavior, and
demonstrates how harvesting the secondary wave
increases the total harvested power collected by the
system.
iLPRC(t)
vpiezo(t)
1
PPRC = ECpiezo f sw = C piezoVPRC 2 f sw ,
2
First
Peak
(1)
where f sw is twice the vibration frequency (due to
rectification of vpiezo(t) by diodes D1-D4) and VPRC is
the maximum piezoelectric voltage at the end of phase
1. Under steady-state conditions, VPRC is given by
Second
Peak
ipiezo(t)
t
VPRC =
−π


1 + e ωτ  ,

2 2
1+ ω τ 

I piezoτ
C piezo
(2)
vpiezo(t)
where τ = C piezo R piezo and ω is the vibration frequency
iLPRC(t)
of the transducer. Combining (1) and (2), the power
from the first peak of the PRC is given as [7]
NSwitch
t
PPRC =
PSwitch
1 23
I piezo 2 R piezo
2π
2
−π

ωτ 
ωτ
1
+
e

 .
2 2
1+ ω τ 

(3)
Occurrence of the 2nd Wave
The occurrence of the secondary wave in the
piezoelectric voltage results from the combination of
the phase shift between vpiezo(t) and ipiezo(t) and the
non-linear switching behavior of the PRC. When
vpiezo(t) is at its maximum amplitude, the PRC extracts
the harvested energy and delivers it to the load almost
instantaneously relative to the vibration period. As a
result of this extraction, vpiezo(t) is reduced to zero
volts. However, immediately following the extraction,
the current source ipiezo(t) is nonzero as shown in Fig.
2. Since the transducer is now operating in phase 1,
ipiezo(t) again charges Cpiezo resulting in the secondary
wave. This secondary voltage waveform can be
described analytically by the equation
Fig. 2: Current and voltage waveforms for energy
harvesting using the PRC.
OPERATION OF THE PRC
Standard operation
The basic operation of the PRC is divided into
three phases, 1-3, as shown in Fig. 2. During phase 1,
both the NSwitch and PSwitch are open, and the
piezoelectric voltage, vpiezo(t), ramps up sinusoidally
as charge accumulates on Cpiezo. Phase 1 is designed to
be much longer than the other two phases, and
therefore the piezoelectric transducer is open-circuited
for a majority of its operation. When vpiezo(t) reaches
its maximum value, corresponding to a maximum
amount of energy stored on Cpiezo (E = ½ CV2), phase
2 begins and NSwitch closes. An LC resonant circuit is
formed between Cpiezo and LPRC, and harvested energy
stored on Cpiezo is transferred to the inductor. The
inductor current, iLPRC(t), rises sinusoidally as vpiezo(t)
falls to zero. Phase 3 begins when all of the energy
has been transferred to LPRC, which is marked by
vpiezo(t) reaching zero and iLPRC(t) reaching its
maximum value. During phase 3, NSwitch is opened and
PSwitch is closed. The energy stored on LPRC is
transferred to the load until iLPRC(t) reaches zero, at
which point PSwitch opens, and the PRC returns to
phase 1.
v piezo ( t ) =
−t


cos (ωt ) − e τ  ,

2 2
1+ ω τ 

I piezoτ
Ceb
(4)
where t=0 is defined at the beginning of the 2nd phase
1. A complete derivation of (4) is presented in
Appendix A.
Unlike the value of the first voltage peak given in
(2) which can be found explicitly, an implicit
numerical solution is required to obtain the peak value
of the secondary wave given in (4).
428
both peaks is greater than that harvested from only the
first peak.
The secondary wave behavior of vpiezo(t) has been
presented for PRC-based energy harvesting systems.
Equations characterizing both the voltage and
harvestable power have been developed and validated
through SPICE simulation. Through the careful design
of transducer parameters, these systems can be
designed for increased performance by utilizing the
second peak. While this work focuses on the
characterization of the secondary voltage peak, it is
possible to imagine a system where additional peaks
(third, fourth, etc…) exist as a result of the PRC
operation. A future goal of this research is to
characterize an optimal design point for a system with
N additional peaks in order to further increase the
total harvested power.
VALIDATION OF THE THEORETICAL
MODEL USING SPICE SIMULATION
SPICE simulation (LTSPICE IV from Linear
Technology) was used to validate the theoretical
models of the secondary voltage wave behavior of
vpiezo(t). Figure 3 shows the secondary peak value of
vpiezo(t), both analytically using (4) and via SPICE
simulation, as a function of Rpiezo and Cpiezo for typical
values encountered in piezoelectric transducers. It can
be seen that higher piezoelectric voltages occur for
lower values of Cpiezo and higher values of Rpiezo. This
behavior can be best explained considering that ipiezo(t)
is fixed for this data, since small values of Cpiezo
charge to higher voltages and large values of Rpiezo
suffer less leakage.
14
Rpiezo = 650 kΩ
10
8
SPICE
Theory
Rpiezo = 550 kΩ
6
Rpiezo = 450 kΩ
Rpiezo = 650 kΩ
5
Rpiezo = 350 kΩ
2nd Peak Power (µW)
2nd Peak Voltage (V)
12
Rpiezo = 250 kΩ
6
4
Rpiezo = 150 kΩ
Rpiezo = 50 kΩ
2
0
0
10
20
Cpiezo (nF)
30
40
50
Rpiezo = 550 kΩ
Rpiezo = 450 kΩ
4
Rpiezo = 350 kΩ
Rpiezo = 250 kΩ
3
Rpiezo = 150 kΩ
Rpiezo = 50 kΩ
2
1
Fig. 3: Maximum voltage values of the second
peak.
0
0
10
20
30
40
Cpiezo (nF)
The maximum power available from the
secondary voltage wave is shown in Fig. 4 using (1)
and the peak voltage of the secondary wave in (4). For
each value of Rpiezo, an optimal value of Cpiezo exists
where the power from the secondary peak is
maximized.
50
．
Fig. 4: Maximum power from the second peak
25
Rpiezo = 650 kΩ
Rpiezo = 550 kΩ
Power (µW)
20
PRC DESIGN USING THE SECOND PEAK
The goal of harvesting energy from the secondary
peak is to increase the total power collected by the
harvesting system. In order to design a system which
does this effectively, it is necessary to understand how
the design parameters affect the total harvested power.
Consider first the effect that harvesting the
secondary peak has on the total power harvested by
the system. Figure 5 compares the total power
obtained from harvesting only the primary peak to the
power produced by harvesting from both the primary
and secondary peaks. An optimal point exists for each
case corresponding to a specific value of the product
of ωτ where the harvested power is maximized
( ωτ =1.76 for the primary only case and ωτ =1.53 for
both). From these results, it can be seen that, at their
respective optimal points, the power harvested from
Rpiezo = 450 kΩ
Power from 1st and 2nd Peak
Power from only 1st Peak
ωτ = 1.53
ωτ = 1.76
Rpiezo = 350 kΩ
15
Rpiezo = 250 kΩ
Rpiezo = 150 kΩ
Rpiezo = 50 kΩ
10
5
0
0
10
20
30
40
50
Cpiezo (nF)
Fig 5: A comparison of power from only the 1st peak
vs. the combined power from 1st and 2nd peaks.
REFERENCES
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2002 Adaptive piezoelectric energy harvesting
circuit for wireless remote power supply IEEE
Trans. Power Electron. 17 669-676
[2] Ottman G, Hoffman H, Lesieutre G, 2003
Optimized piezoelectric energy harvesting
circuit
using step-down converter
in
429
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[7]
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Extraction Proceedings of ASME International
−t
v piezo ( t ) = ke τ + A cos (ωt + φ ) + B sin (ωt + φ ) , (6)
where k is an integration constant, and A and B are
A=−
B=
I piezoωτ 2
C piezo (1 + ω 2τ 2 )
I piezoτ
.
(7)
C piezo (1 + ω 2τ 2 )
Assuming that the PRC is operating in steady-state
and the waveforms in Fig. 2 are periodic, the
beginning of the second wave can be designated
as t = 0 . The integration constant k is then found to
be
k=
I piezoτ
C piezo (1 + ω 2τ 2 )
ωτ cos (φ ) − sin (φ ) 
(8)
using the initial condition
v piezo ( 0 ) = 0 .
Mechanical Engineering Congress and Exposition
2006 (Chicago, IL 5-10 November 2006)
(9)
The total solution of vpiezo(t) then becomes
APPENDIX A
Derivation of vpiezo(t) during the secondary wave
After the occurrence of the first peak and the
associated extraction of energy, both the NSwitch and
PSwitch are off, and the transducer operates in open
circuit mode. Using Kirchoff’s Current Law, the
behaviour of the transducer can be described by
I piezo sin (ωt + φ ) =
v piezo ( t )
R
+C
dv piezo ( t )
dt
,
v piezo ( t ) =
I piezoτ
Ceb (1 + ω 2τ 2 )
−
 ωτ cos (ωt + φ )
. (10)
+ sin (ωt + φ ) + (ωτ cos (φ ) − sin (φ ) ) e − t /τ 
Substituting in the value of φ and simplifying terms,
it can be found that for the secondary wave
(5)
v piezo ( t ) =
where φ = π / 2 + tan −1 (ωτ ) is the phase shift
between vpiezo(t) and ipiezo(t). The total solution to the
differential equation in (5) is the sum of the
homogeneous and particular solutions given by
430
−t


 cos (ωt ) − e τ  .
2 2
1+ ω τ 

I piezoτ
Ceb
(11)